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Theorem lineval2b 26189
Description: The point  B belongs to the line passing through it . (For my private use only. Don't use.) (Contributed by FL, 25-Feb-2016.)
Hypotheses
Ref Expression
lineval2a.1  |-  P  =  (PPoints `  G )
lineval2a.3  |-  M  =  ( line `  G
)
lineval2a.4  |-  ( ph  ->  G  e. Ig )
lineval2a.5  |-  ( ph  ->  A  e.  P )
lineval2a.6  |-  ( ph  ->  B  e.  P )
Assertion
Ref Expression
lineval2b  |-  ( ph  ->  B  e.  ( A M B ) )

Proof of Theorem lineval2b
StepHypRef Expression
1 lineval2a.1 . . . 4  |-  P  =  (PPoints `  G )
2 lineval2a.3 . . . 4  |-  M  =  ( line `  G
)
3 lineval2a.4 . . . 4  |-  ( ph  ->  G  e. Ig )
4 lineval2a.6 . . . 4  |-  ( ph  ->  B  e.  P )
51, 2, 3, 4, 4lineval2a 26188 . . 3  |-  ( ph  ->  B  e.  ( B M B ) )
6 oveq1 5881 . . . 4  |-  ( A  =  B  ->  ( A M B )  =  ( B M B ) )
76eleq2d 2363 . . 3  |-  ( A  =  B  ->  ( B  e.  ( A M B )  <->  B  e.  ( B M B ) ) )
85, 7syl5ibr 212 . 2  |-  ( A  =  B  ->  ( ph  ->  B  e.  ( A M B ) ) )
9 eqid 2296 . . . . 5  |-  (PLines `  G )  =  (PLines `  G )
103adantl 452 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  G  e. Ig )
11 lineval2a.5 . . . . . 6  |-  ( ph  ->  A  e.  P )
1211adantl 452 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  A  e.  P )
134adantl 452 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  B  e.  P )
14 simpl 443 . . . . 5  |-  ( ( A  =/=  B  /\  ph )  ->  A  =/=  B )
151, 9, 2, 10, 12, 13, 14lineval22 26185 . . . 4  |-  ( ( A  =/=  B  /\  ph )  ->  ( A  e.  ( A M B )  /\  B  e.  ( A M B ) ) )
1615simprd 449 . . 3  |-  ( ( A  =/=  B  /\  ph )  ->  B  e.  ( A M B ) )
1716ex 423 . 2  |-  ( A  =/=  B  ->  ( ph  ->  B  e.  ( A M B ) ) )
188, 17pm2.61ine 2535 1  |-  ( ph  ->  B  e.  ( A M B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874  PPointscpoints 26159  PLinescplines 26161  Igcig 26163   linecline 26179
This theorem is referenced by:  lineval5a  26191  isibg1a8  26230  segline  26244  lppotos  26247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-ig2 26164  df-li 26180
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